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Thursday, January 12, 2017

Thermodynamics Problems with Solutions Two

We
are solving series of problems on the topic thermodynamics. We need to be
familiar with the concepts like specific heat and latent heat. Specific heat is
the amount of heat energy required to raise the temperature of unit mass of
substance by one degree centigrade or kelvin. In the case of gases, we get two
different kinds of specific heats one at constant pressure and the other at
constant volume. Latent heat is the amount of heat energy required to convert
unit mass of substance from one state to other and it happens at constant
temperature. As the heat energy is also conserved, the heat lost by a hot body
is the heat gained by the cold body when the bodies are connected and there is
no loss of heat energy in any format.

Problem

The
ratio of specific heats of has is given to us in the problem. We need to
measure the change in the internal energy of the system when volume gets
doubled at constant pressure. Problem is as shown in the diagram below.

Solution

We
can write the value of internal energy in terms of first law of thermodynamics
and work done can be written as the
product of pressure and change in volume of the system. It can be further
simplified as shown in the diagram below.

Problem

One
mole of mono atomic gas is mixed with one mole of diatomic gas are mixed. We
need to find the effective specific heat of the system and the problem is as
shown in the diagram below.

Solution

We
can write the equation for the effective specific heat by taking the
conservation of energy into consideration as shown in the diagram below. We
need to apply the value of specific heat of mono and diatomic gas values as per
the standards and solve the problem as shown below.

Problem

Certain
amount of heat is required to raise the temperature to a certain value at
constant volume and we need to find out the heat energy required to raise the
temperature of the system at constant pressure and the problem is as shown in
the diagram below.

Solution

When
the volume is constant, to find the heat energy required to raise the
temperature to a certain value, we need to use the definition of specific heat
of gas at constant volume. To find the heat required to raise the temperature
of the system at constant pressure, we need to use the definition of specific
heat of gas at constant pressure. The detailed solution is as given in the
diagram below.

Problem

One
mole of ideal gas expands to double to its volume under isothermal conditions
at a given constant temperature as shown in the diagram below. We need to
measure the work done in this case.

Solution

We
know that isothermal process means the work is done at constant temperature.
Here pressure is not constant and to measure the work done, we shall integrate
the pressure impact under the volume conditions and we can derive a mathematical
equation as we have learned. Taking that into consideration, we can solve the
problem as shown in the diagram below.

Problem

For
an adiabatic expansion, if a monoatomic gas expands by a certain percentage, we
need to measure the percentage variation in its pressure and the problem is as
shown in the diagram below.

Solution

We
know that in adiabatic process, the heat energy of the system is constant and
it is not going to change. Here temperature is not constant and hence Boyle’s
law is not valid here. The relation between pressure and volume are different
and taking that into consideration, we can solve the problem as shown in the
diagram below.